In 1974, Manakov first carried out the study on the long-time behavior of nonlinear wave equations solvable by the inverse scattering method [1]}.
Then by using this method, Zakharov and Manakov give the first result for large-time asymptotic of solutions for the NLS equation with decaying initial value in 1976 [2]}. The inverse scattering method also worked for long-time behavior of integrable systems such as KdV, Landau-Lifshitz and the reduced Maxwell-Bloch system [3]}, [4]}, [5]}. In 1993,
Deift and Zhou developed a nonlinear steepest descent method to rigorously obtain the long-time asymptotics behavior of the solution for the MKdV equation
by deforming contours to reduce the original Riemann-Hilbert problem (RHP) to a model one whose solution is calculated in terms of parabolic cylinder functions [6]}.
Since then this method
has been widely applied to the focusing NLS equation, KdV equation, Fokas-Lenells equation, derivative NLS equation, short-pluse equation and Camassa-Holm equation etc. [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}.
